Two generators have cost functions with incremental-cost characteristics:
\[ \frac{dF_1}{dP_1} = 40 + 0.2 P_1, \frac{dF_2}{dP_2} = 32 + 0.4 P_2 \] They must supply a total load of 260 MW. Find the optimal generation (economic dispatch) ignoring losses.
Two generators have cost functions with incremental-cost characteristics:
\[ \frac{dF_1}{dP_1} = 40 + 0.2 P_1, \frac{dF_2}{dP_2} = 32 + 0.4 P_2 \] They must supply a total load of 260 MW. Find the optimal generation (economic dispatch) ignoring losses.
Show Hint
- \(P_1 = P_2 = 130\)
- \(P_1 = 160,\; P_2 = 100\)
- \(P_1 = 140,\; P_2 = 120\)
- \(P_1 = 120,\; P_2 = 140\)
The Correct Option is B
Solution and Explanation
Step 1: Economic dispatch condition.
For optimal operation (no losses), incremental costs must be equal:
\[
40 + 0.2P_1 = 32 + 0.4P_2
\]
Step 2: Use total power constraint.
\[
P_1 + P_2 = 260
\]
Step 3: Solve the system.
Subtract constants:
\[
40 - 32 = 0.4P_2 - 0.2P_1
\]
\[
8 = 0.4P_2 - 0.2P_1
\]
Substitute \(P_2 = 260 - P_1\):
\[
8 = 0.4(260 - P_1) - 0.2P_1
\]
\[
8 = 104 - 0.4P_1 - 0.2P_1
\]
\[
0.6P_1 = 96
\]
\[
P_1 = 160, P_2 = 100
\]
Step 4: Conclusion.
Only option (B) satisfies both economic dispatch and total load constraints.
Final Answer: \(\boxed{P_1 = 160,\; P_2 = 100}\)
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